Neighborly measures for sampling independent sets, and a perturbative approach to the question of uniqueness

Recently, there has been great interest in understanding the fundamental
limits of our ability to sample from the independent sets (i.s.) of a
graph. One approach involves the study of the so-called hardcore model,
in which each i.s. is selected with probability proportional to some fixed
activity $\lambda$ raised to the cardinality of the given i.s. It is
well-known that for any fixed degree $\Delta$, there exists a critical
activity $\lambda_{\Delta}$ s.t. for all activities below
$\lambda_{\Delta}$, the sampled i.s. enjoys a long-range independence
(a.k.a. uniqueness) property when implemented on graphs with maximum
degree $\Delta$, while for all activities above $\lambda_{\Delta}$, the
sampled i.s. exhibits long-range dependencies. Such phase transitions are
known to have deep connections to the inherent computational complexity of
the underlying combinatorial problems. In this talk, we study a family of
measures which generalizes the hardcore model by taking more structural
information into account, beyond just the number of nodes belonging to the
i.s., with the hope of further probing the fundamental limits of what we
can learn about the i.s. of a graph using only local information. In our
model, the probability assigned to a given i.s. depends not only on its
cardinality, but also on how many excluded nodes are adjacent to exactly
$k$ nodes belonging to the i.s., for each $k$, resulting in a parameter
for each $k$. We generalize the notion of critical activity to these
``neighborly measures", and give necessary and sufficient conditions for
long-range independence when certain parameters satisfy a
log-convexity(concavity) requirement. To better understand the phase
transitions in this richer model, we view the classical critical activity
as a particular point in the parameter space, and ask which directions can
one move and still maintain long-range independence. We show that the set
of all such ``directions of uniqueness” has a simple polyhedral
description, which we use to study how moving along these directions
changes the probabilities associated with the sampled i.s. We conclude by
discussing implications for choosing how to sample when trying to optimize
a linear function of the underlying probabilities.